On Conjecture (Unsourced)

"In the study of Nature conjecture must be entirely put aside, and vague hypothesis carefully guarded against. The study of Nature begins with facts, ascends to laws, and raises itself, as far as the limits of man’s intellect will permit, to the knowledge of causes, by the threefold means of observation, experiment and logical deduction." (Jean Baptiste-Andre Dumas)

"Indeed, when in the course of a mathematical investigation we encounter a problem or conjecture a theorem, our minds will not rest until the problem is exhaustively solved and the theorem rigorously proved; or else, until we have found the reasons which made success impossible and, hence, failure unavoidable. Thus, the proofs of the impossibility of certain solutions plays a predominant role in modern mathematics; the search for an answer to such questions has often led to the discovery of newer and more fruitful fields of endeavour." (David Hilbert)

"The conjectures of the scientific intelligence are genuine creative novelties, inherently unpredictable and not determined by the character of the scientist’s physical environment. The thinking mind is not a causal mechanism." (Anthony M Quinton)

"The only use of an hypothesis is, that it should lead to experiments; that it should be a guide to facts. In this application, conjectures are always of use. The destruction of an error hardly ever takes place without the discovery of truth. [...] Hypothesis should be considered merely an intellectual instrument of discovery, which at any time may be relinquished for a better instrument. It should never be spoken of as truth; its highest praise is verisimility. Knowledge can only be acquired by the senses; nature has an archetype in the human imagination; her empire is given only to industry and action, guided and governed by experience." (Sir Humphry Davy) 

"The purpose of life is to conjecture and prove." (Paul Erdős)

"The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations." (Godfrey H Hardy)

"What certainty can there be in a Philosophy which consists in as many Hypotheses as there are Phaenomena to be explained. To explain all nature is too difficult a task for any one man or even for any one age. 'Tis much better to do a little with certainty, & leave the rest for others that come after you, than to explain all things by conjecture without making sure of any thing." (Sir Isaac Newton)